In many engineering applications, it is a formidable task to construct mathematical modelsudthat are expected to produce accurate predictions of the behavior of a system of interest.udDuring the construction of such predictive models, errors due to imperfect modeling anduduncertainties due to incomplete information about the system and its environment (e.g.,udinput or excitation) always exist and can be accounted for appropriately by usingudprobability logic. To assess the system performance subjected to dynamic excitations, audstochastic system analysis considering all the uncertainties involved has to be performed. Inudengineering, evaluating the robust failure probability (or its complement, robust reliability)udof the system is a very important part of such stochastic system analysis. The word ‘robust’udis used because all uncertainties, including those due to modeling of the system, are takenudinto account during the system analysis, while the word ‘failure’ is used to refer toudunacceptable behavior or unsatisfactory performance of the system output(s). Wheneverudpossible, the system (or subsystem) output (or maybe input as well) should be measured toudupdate models for the system so that a more robust evaluation of the system performanceudcan be obtained. In this thesis, the focus is on stochastic system analysis, model andudreliability updating of complex systems, with special attention to complex dynamic systemsudwhich can have high-dimensional uncertainties, which are known to be a very challengingudproblem. Here, full Bayesian model updating approach is adopted to provide a robust andudrigorous framework for these applications due to its ability to characterize modelinguduncertainties associated with the underlying system and to its exclusive foundation on theudprobability axioms. First, model updating of a complex system which can have high-dimensional uncertaintiesudwithin a stochastic system model class is considered. To solve the challengingudcomputational problems, stochastic simulation methods, which are reliable and robust toudproblem complexity, are proposed. The Hybrid Monte Carlo method is investigated and itudis shown how this method can be used to solve Bayesian model updating problems ofudcomplex dynamic systems involving high-dimensional uncertainties. New formulae forudMarkov Chain convergence assessment are derived. Advanced hybrid Markov ChainudMonte Carlo simulation algorithms are also presented in the end.udNext, the problem of how to select the most plausible model class from a set of competingudcandidate model classes for the system and how to obtain robust predictions from theseudmodel classes rigorously, based on data, is considered. To tackle this problem, Bayesianudmodel class selection and averaging may be used, which is based on the posteriorudprobability of different candidate classes for a system. However, these require calculationudof the evidence of the model class based on the system data, which requires theudcomputation of a multi-dimensional integral involving the product of the likelihood andudprior defined by the model class. Methods for solving the computationally challengingudproblem of evidence calculation are reviewed and new methods using posterior samples areudpresented.udMultiple stochastic model classes can be created even there is only one embeddeduddeterministic model. These model classes can be viewed as a generalization of theudstochastic models considered in Kalman filtering to include uncertainties in the parametersudcharacterizing the stochastic models. State-of-the-art algorithms are used to solve theudchallenging computational problems resulting from these extended model classes. Bayesianudmodel class selection is used to evaluate the posterior probability of an extended modeludclasse and the original one to allow a data-based comparison. The problem of calculatingudrobust system reliability is also addressed. The importance and effectiveness of theudproposed method is illustrated with examples for robust reliability updating of structural systems. Another significance of this work is to show the sensitivity of the results ofudstochastic analysis, especially the robust system reliability, to how the uncertainties areudhandled, which is often ignored in past studies.udA model validation problem is then considered where a series of experiments are conductedudthat involve collecting data from successively more complex subsystems and these data areudto be used to predict the response of a related more complex system. A novel methodologyudbased on Bayesian updating of hierarchical stochastic system model classes using suchudexperimental data is proposed for uncertainty quantification and propagation, modeludvalidation, and robust prediction of the response of the target system. Recently-developedudstochastic simulation methods are used to solve the computational problems involved.udFinally, a novel approach based on stochastic simulation methods is developed usingudcurrent system data, to update the robust failure probability of a dynamic system which willudbe subjected to future uncertain dynamic excitations. Another problem of interest is toudcalculate the robust failure probability of a dynamic system during the time when theudsystem is subjected to dynamic excitation, based on real-time measurements of some outputudfrom the system (with or without corresponding input data) and allowing for modelinguduncertainties; this generalizes Kalman filtering to uncertain nonlinear dynamic systems. Forudthis purpose, a novel approach is introduced based on stochastic simulation methods toudupdate the reliability of a nonlinear dynamic system, potentially in real time if theudcalculations can be performed fast enough.
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